Contributions

to the

Theory of

Life Insurance Reserves

by

Dr. August Zillmer
Actuary, Germania Stock Company
Stettin, Prussia

Stettin, 1863

Press of Theodore von der Nahmer
Translated by
Bill Roach and Gunnar Alksnis
Spring, 1989 Copyright


 
 



Translators' Note


All of the financial figures of the original paper have been reproduced in the translation. This has been achieved by substituting dollars for the Prussian taler. The units of Prussian currency at the time were as follows:


A single taler in 1861 was worth $.75 in U.S. currency. Reconstructions of the Consumer Price Index for that era show a CPI of 27 for 1861 where the CPI of 1967 is 100. All of the tables in the origina paper have been recalculated using the spreadsheet software Quattro. Except where indicated, the only differences between the recalculated values and the original values are due to rounding.
 



 
 
 

Introduction
 

More than any other corporations life insurance companies should be held to exact and strict accounting rules. The entry into a life insurance contract is an act of self denial and an example of the noblest love of spouse or child. Often the policy is the only inheritance and the total estate of widows and orphans. Since life insurance institutions draw in a considerable portion of the savings of thousands of policy holders on the basis of contracts several of which will not expire before a half or sometimes almost a whole century. Therefore, their solidity and solvency must be mathematically provable and indisputable.

These undeniably true words which were spoken by the New York State Insurance Commissioner when he withdrew the license of the American Mutual Life Insurance Company of New Haven, Connecticut, to do business in the State of New York. In honor of these words, in the following pages we will present several considerations about two different methods of calculating the reserves of life insurance institutions.
 

In the year 1857, Mr. Wilhelm Lazarus , in a thorough article (See Rundschau der Versicherungen, 1857, p317.) had already called attention to the misleading principle of estimating the present value of obligations at a lower interest rate than the present value of the yet to be collected premiums and demonstrated the consequences of such misleading calculations on the balance sheet of a company which thus calculates. Yet since that time, practically nothing has happened to prevent the use of this misleading principle. Here and there, one finds in professional journals the remark that the reserve figured on gross premiums is too small, but that is all one finds. In more recent times, prompted by the publication of the balance sheets of some English companies, some voices have been raised against the mischief of these calculations. With this, other aspects of the life insurance business have been drawn into the conversation and, here and there, mistaken concepts have become widely shared. For example, it was asserted that the recent custom of paying agents a high commission for the acquisition of a life insurance contract, about 1 % of the insured amount, seems to make it impossible to set up an actuarially sound reserve. And thus companies that pay such commissions are forced to calculate reserves too low, if not even according to the principle of gross premiums. The presentation of such mistaken, when not intentionally misleading assertions, are well suited to deeply affect the trust of the public in life insurance companies and will harm exactly those companies that have published their business details with the greatest openness and make no secret of their commissions. We attempt to contribute, to the best of our ability, the correct view about the reserves of life insurance companies and in the first segment of the following monograph provide the proof that even for the custom of a closing commission, a reserve can be established that has a completely calculable magnitude. In the second segment, we will briefly discuss the method of calculation with gross premiums, and we believe that we thereby show that many phenomena existing in the real world are natural consequences of this misleading system.
 
 



 
 


I.


In a life insurance institution, the reserve plays a large role. The annual premium for an insurance policy [in this we have in mind the usual whole life insurance policy with annual premium payment] remains constant for the whole term of the insurance policy while the risk that the insured amount becomes due by the death of the insured grows with each year. And if an insurance policy were always issued for one year only, it would require an increase in premium each year. Therefore a premium that remains constant for the whole term of the insurance policy is greater for the first year after the issue of the insurance policy than the yearly renewable term (YRT) premium; later the premium is smaller than the YRT premium. If a certain number of persons enter an insurance bank, then the bank will create a considerable surplus after covering the occurring cases of death among those persons. And in the later years this surplus has to cover the added expenditures when the bank must pay more for cases of death than it receives in premiums. For example, the attached Table II shows that a bank which insures a number of 30 year old persons with a sum of $500,000, collects more in premium during the first 25 years than it has to pay out for cases of death. From the 26th year on, the expenses for deaths overtake the premium income. For example, in the 40th year, the excess of the death claims over the premiums is $8,679.
 

Therefore as the compelling consequence of the level premium method, a life insurance bank must set up a reserve fund. In each case, the amount of the reserve fund cannot be arbitrarily set as it suits the bank, but it has to be mathematically and precisely determined, just as the annual premium is calculated mathematically and precisely on the basis of a mortality table and a certain rate of interest. For example, the insurance bank may not lower the reserve fund, speculating that the mortality among the insured will be more advantageous than could be expected from the mortality table used in the calculation of the premium or that it will, in consequence of fortunate speculations, obtain a higher interest income with their monies and thus be easily able to cover the shortage in the reserve funds. The reserve fund must be set up much more from the paid in premiums of the existing insurance policies and the calculable interest after deduction of the same calculable future expenses for death, and to be sure the calculable expenses, not the actually caused ones for otherwise a sustained excess mortality among the insureds could, if not exhaust the reserve fund, at least significantly lower it below the level required for the future performance of the bank. The reserve fund is not to be set up from the premiums paid by the insured but from the net premiums. The premiums arrived at by exact calculation (net premiums) are to be increased by a loading which should reimburse the bank for business expenses and the risk that mortality among the insureds exceeds the expected limits. The loading, the size of which will be established in the individual banks according to more or less arbitrary rules of thumb, goes to the bank, and the reserve fund is calculated from the remaining net premiums after the deduction of this loading. As soon as the net premium described above is determined, the reserve will result as a mathematical consequence at every point of time during the term of the insurance policy.
 

As a rule the net premium is so established that it remains constant for the whole term of the insurance policy, just as the loaded premium paid by the insured remains constant. But clearly there is no reason not to include other considerations relating to the amount of net premium if only these are rationally calculated, completely covering the obligations of the bank. There are insurance policies with rising or falling premium payments whereby naturally not only the loaded premium changes in the different periods during the term of the insurance policy, but also the net premium.
 

To arrive at that firmly established net premium that we believe we must prefer in general and prefer especially in the current situation in the business, we put forward several considerations:

The participation of the German public in life insurance has increased in a gratifying fashion in recent years. Not only do some of the few older German companies do more significant business than in earlier decades, but also the newer German companies and besides them a considerable number of foreign companies work with generally growing success. The Germania has even experienced such a spectacular participation by the public that last year (1862) it issued insurance contracts in a sum of significantly more than $6 million and in the month April of this year alone, in a sum of more than $1 million. This glorious upswing in the life insurance business is not only the consequence of the correct appreciation and high significance which life insurance has from the individual, the family, and the state and which is permeating all levels of society, but this upswing is also brought about by the efforts of the executives of the life insurance companies and their agents. These executives introduced the practice of paying the agent a higher commission at the issue of the insurance policy [previously the agent had only received a commission which remained constant for the term of the insurance policy. The agent kept a portion of the premiums he collected and thus achieved for himself a worthwhile income only after years and years]. This succeeded in attracting active business men as agents who then, being placed in a position to quit their other business, were able to use their whole strength for the attraction of new insurance policies. The success, that the introduction of this commission practice has had until now is its own best justification. But complaints against the commission method force another question: how does a company defray the agency costs which must grow proportionately for a large scale addition of new insurance policies. We admit that if a company is required not only to post an actuarially sound reserve [naturally this must be required] but if it is also required to set the amount of the reserve on the basis of the determined, prescribed, or traditional calculation model (net level premium ), then company can thereby run into difficulties. The expenses of an insurance policy, for the coverable mortality for the first year and for the reserve set aside at the end of the first year according to any one calculation model can and must under all circumstances be greater than the premium income for this insurance policy. The thus growing deficit will naturally be the greater, the greater the increase in new insurance policies is, and when the increase is sufficiently large, the deficit too will become so large that even a company with significant means could run into difficulties. It should be clear even to the lay person that a contradiction emerges here. On the one hand, one wishes ample increases in new insurance policies; on the other hand, one would be financially embarrassed by this and this could happen even [as will be demonstrated in the following] with commission rates that have not been set too high.

It would be wonderful if a solution for this could not only be found but if it offered itself, and this is really the case. The obvious thought is that under the old, usual method of calculation which at the close of an insurance policy posts the costs growing out of that closing, possibly in a prepaid commission account, and this account is entered as an asset of the bank. The sums brought into this account in an individual year would then have to be amortized in a certain number of years, or what would be preferable is that from the total amount of the prepaid commission each year a certain and adequate portion of the premium income would be written off. This model was publicly discussed on April 26 of this year in an article in the German insurance paper of Breslau, but for various reasons we cannot sanction this. For one, there is no measure and no limit for the amount of those costs that could be placed in this prepaid commission account. In essence this account would have the effect of lessening the base capital or paid in the reserve; and then it would certainly be possible to endanger the stability of the company through excessive expenses for the acquisition of new insurance policies. Second, the artificial asset of prepaid commissions should really be gradually amortized. Since this requires a relatively high fraction of the annual premium, this means that the coverage of the other costs and the gain for a long series of years, respectively, are unnecessarily lowered. Finally, thirdly, there could and would be found, in the account of the prepaid commissions, sums prepaid for such insurance policies that have lapsed. Later insureds then have to pay subsequently the costs which have been incurred by earlier insureds.

In the following simple way one develops a model which is completely untouched by these disadvantages. One calculates the net premium for an insurance policy in such a way that in the first year it is smaller by a certain amount than for the following years. We omit the development of the necessary formula here. It is, by the way, extremely simple. We get our net premium as the usual premium which remains constant for the whole term of the insurance policy. Take the amount by which the first annual premium should be smaller and divide it by an annuity due for the issue age and add it to the usual net premium.



 
 



For example, calculating with 3 1/2 percent interest and using the Combined Experience Table, the net annual premium at issue age 40 for an endowment for $100 at age 90 [and in the following, this is what we always have in mind] is 2.484 percent. Should the first annual premium be smaller by 1 percent of the insured sum, then we get as the premium for the following years 2.484 + 1 divided by the annuity due at age 40 which ends at 90, that is, 2.484 + 0.059 and this as premium for the first year 2.543 - 1. = 1.543. In contrast, if the first premium should be smaller by 1 1/4 percent of the insured sum, then the first premium is 2.484 + 5/4 * 0.059 - 1.25 = 1.308 and the premium for all the following years is 2.484 + 5/4 * 0.059 = 2.558.
 

Naturally also the insured pays the same premium in the first year which he will have to remit in the following years, and the bank covers the cost arising from acquisition of the insurance policy with the portion of the premium available to it in the first year. The reserve will be calculated just as with the usual premium, that is, one finds the reserve for an insurance policy which has been in existence for a number of years by deducting from the net single premium at the attained age the product of



 
  While the formulas are developed prospectively, Zillmer's reserve calculations are actually done retrospectively as was common in the 19th century.


 
 

the net annual premium and an annuity due at the higher age. The reserve will be somewhat smaller here than according to the previous method, but exactly fulfills the demands:

  1. that it is formed from the net premiums, after one takes deductions of the calculable expenses for death, and with the addition of the calculable interest, or
  2. in other words, that the reserves in union with the premium to be paid by the insured in the future and, that is of course, with the net premiums, completely covered the obligations of the company. Since the reserves are somewhat smaller than according to the old (net level premium) method, the net premiums are by a small, but corresponding, part higher.
Before we enter into further considerations of the reserve, we want to say something more about the first year premium. For this premium, for each issue age, there exists a minimum, and this is given by the premium for one year of term insurance. If the premium for the first year were much smaller than the corresponding premium for one year of term insurance, then the mortality of the first year would not be completely covered by the premium, and the insurance company would have to cover the excess mortality out of other means which under circumstances could become not only uncomfortable, but even dangerous. Insofar as the minimum first year premium is given by the premium for one year of term insurance, the maximum of the difference between the premium for subsequent years and the premium for the first year is given at the same time for every age. If we call this difference X, then obviously the usual premium increased by X divided by the annuity due at the issue age must be just as large as the premium for one year of term insurance increased by X.



 

If we solve the resulting equation, we find X, that is, the maximum of the costs that may be incurred at the closing of an insurance policy equal to the difference between the usual net premium and the premium for one year of term insurance, this difference multiplied by the annuity due at the issue age and divided by the annuity due lessened by one or immediate annuity at the same issue age.

The formula for the maximum of the closing costs can be expressed in many ways. So this maximum, for example, is equal to the quotient of the annuity due at the issue age lessened by one and the annuity of the next higher age lessened by one, that is, for the insurance sum one, or it is equal to the difference of the usual net premium at a one year higher age and the usual net premium of the issue age multiplied by the annuity due at the issue age or, and this is the simplest expression, the maximum of the closing costs is equal to the net premium for the one year higher age lessened by the premium of the issue age for a one year term insurance policy. This simple form for the maximum of the closing costs can also be directly deduced. The maximum that the company can spend for the acquisition of an insurance policy is evidently reached when at the end of the first year, the reserve is zero. The latter is the case if we take as net premium the usual net premium of the next higher age. Since the premium for the one year term insurance policy is used to cover mortality, there remains for the coverage of costs caused by closing at most the difference between the usual net premium of the next higher age and the premium for the one year term insurance policy.
 

The following table shows the maximal rates for the closing costs if one calculates according to the Combined Experience Table and 3 1/2 percent interest.
 
 
 
 
Issue  Maximum Rate of Closing Cost  Maximum Rate of Closing Cost 
Age x Percent of Insured Sum Percent of Premium
10 0.48  42.4% 
15 0.59  46.7%
20 0.71  50.1%
25 0.86  53.3%
30 1.04  56.0%
35 1.27  58.5%
40 1.58 61.1%
45 1.95  62.3%
50 2.33  60.2%
55 2.77  57.0%
60 3.28  52.8%
65 3.80  47.2%




 
 

It follows from this table that if the average issue age is 37 years or a little more, closing costs of a little more than 1 3/8 percent of the insured sum can be applied; if we take into consideration that the maximum rate for higher ages grows in an increasing relationship and that furthermore the younger issue ages purchase smaller sums of insurance, then we may assert that an insurance company which figures according to the Combined Experience Table and an interest rate of 3 1/2 percent can establish 1 1/2 percent of the insured sum as the average maximum that it may spend for the acquisition of an insurance policy and that it may cover by a corresponding calculable portion of the net premium.
 

It does not follow from this that 1 1/2 percent of the insured sum has to be figured as the closing cost; on the contrary, I hold it advisable to stay with 1 percent or 1 1/4 percent of the insured sum. It is possible for the average issue age to decrease if younger persons participate in insurance in greater numbers or if younger persons insure themselves with larger amounts than the present experience has shown. Then it is possible for the results of the calculation to develop so that the average issue age decreases and the calculable rate of closing costs has become too high.
 

Since even in the case where we calculate the closing costs with only 1 - 1 1/4 percent of the insured sum, with high probability but not with absolute certainty we are secured against the danger that could emerge from a decline in the average issue age; therefore I consider a precautionary measure necessary which will be discussed in the following:

If we figure as closing costs more than the maximum rate for the youngest issue age, then at the younger ages the net premium to be used for the first year for the risk is smaller than the premium for a one year term insurance policy at the issue age or, in other words, the reserve at the end of the first insurance year becomes negative and because of this the net premium to be used in following years would be greater than the usual net premium for an issue age one year higher. If the company includes the negative reserve for a younger age in the total sum of the reserves this lowers the total reserves exactly by as much as the company has too little for the calculable mortality in the course of the year [of course, drawing interest until the end of the year]. The higher the closing costs are calculated, the greater will be the amount of the negative reserves and by so much more would the total sum of the reserves be forced down thereby. Add to this the fact that the cost must be written off against the company's capital for those insurance policies which lapse after a 1 year existence. In consideration of these circumstances, I hold it essential that a company does not include the negative reserves in the total reserves, but calculates the reserve for the end of the year as zero. From this grows for the company an expense which exceeds the calculable expense allowance. But this is easy to handle, as will be demonstrated, so long as we can assume that the calculable closing costs have not been set too high. If the conditions change so that it appears that the closing costs calculated up to now have been set too high, this will become evident when the portion of income that would be used for the formation of the profit is reduced while the premium reserve remains untouched and the company has, at the same time, a built in regulator for the magnitude of the closing costs.

The insurance polices that had a negative reserve at the end of the first year are increasing more in the second year because of the consideration of the calculable closing costs. If at the end of the second year, the reserve should still be negative then instead of the negative value again we set the reserve to zero and pay the calculable reserve only at the end of the third year. This last happened only with the very lowest issue ages as the following table shows:
 
 
 
 
Issue Net Level Premium Reserves
Age x for a Policy for $100
Durations 
1 2 3
15 0.584  1.187  1.809 
16 0.606  1.232  1.878
17 0.630  1.279  1.949

 
 
Zillmer Reserves @ 1 1/4 % Closing Costs
for a Policy for $100
Durations 
1 2 3
15 -0.659  -0.049 0.581 
16 -0.636  -0.002  0.651 
17 0.612  0.045  0.723





 
 

In general, the expense that the bank incurs by not setting up the actual negative reserves but setting them to zero, that is, for some of the insurance policies the reserves are calculated as greater than they actually are, is not significant. If we calculate on the basis of 1 1/4 percent closing cost, the reserves will already be positive at the end of the first year for an issue age of 35 years; if we calculate on the basis of 1 percent, the reserves will already be positive for the issue age of 30. In general, the younger ages are covered with less insurance the lower the age so that where the closing expense for the individual insurance policy becomes somewhat larger, then there are only individual insurance policies that cause this expense. Add to this that in a steadily growing business, this expense is to be paid only in the first year. The net premium for the second year has to cover the mortality of that year and the negative reserve for the end of the first year and beyond that provide the positive reserve for the end of the second year. Since the negative reserve has already been defrayed by the bank, this portion of the net premium flows towards its profit or rather this portion defrays the negative reserves for the new insurance policies. In a business that grows at an increasing rate, the expense would repeat itself every year, and the increase would be the difference by which the insurance policies for the younger ages have grown more than in the previous year. To have grasp of the size of these costs we will make the specific and certainly not too low assumption that in a year bank insures

$10,000 for 20 year old persons
$20,000 for 21 year old persons
$30,000 for 22 year old persons
...
$100,000 for 29 year old persons
$120,000 for 30 year old persons
$140,000 for 31 year old persons
...
$200,000 for 34 year old persons.
 
Then if we calculate closing costs as 1 percent of the insured amounts, we would have
 
 
 
Negative Reserve 
Issue Age Insured Sum as Percent Insured Sum in Total
20 
$10,000 
-0.289 
-28.9 
21 
$20,000 
-0.263 
-52.5 
22 
$30,000 
-0.234 
-70.3 
23 
$40,000 
-0.205 
-82.2 
24 
$50,000 
-0.175 
-87.5 
25 
$60,000 
-0.143 
-85.8 
26 
$70,000 
-0.110 
-77.1 
27 
$80,000 
-0.075 
-60.4 
28 
$90,000 
-0.040 
-35.9 
29 
$100,000 
-0.002 
-2.2 
Sum 
$550,000 
 
-582.8 



If we calculate closing costs as 1 1/4 percent of the insured amount
 
 
 
 
 
1 1/4 % Closing Cost 
Negative Reserve 
Issue Age Insured Sum As Percent Insured Amount In Total
20 
$10,000 
-0.538 
-53.8 
21 
$20,000 
-0.511 
-102.1 
22 
$30,000 
-0.482 
-144.7 
23 
$40,000 
-0.453 
-181.4 
24 
$50,000 
-0.423 
-211.5 
25 
$50,000 
-0.391 
-234.5 
26 
$60,000 
-0.358 
-250.6 
27 
$70,000 
-0.323 
-258.6 
28 
$80,000 
-0.288 
-258.8 
29 
$90,000 
-0.250 
-249.7 
30 
$100,000 
-0.211 
-253.0 
31 
$120,000 
-0.170 
-237.4 
32 
$140,000 
-0.127 
-203.1 
33 
$160,000 
-0.082 
-147.0 
34 
$180,000 
-0.033 
-66.8 
Sum 
$1,350,000 
 
-2853.0 

 

If a company in every year issues $550,000 more insurance inforce than in the preceding year for the issue ages under 30, then it would have, at 1 percent closing costs, an expenditure that exceeds the calculable only by $586, that is, this expenditure does not even amount to 1/4 percent of the premium income as soon as the premium income has climbed over $250,000, or if the company figures on 1 1/4 percent closing costs, it would have an annual expenditure of $2,870 when in every year it issues policies for $1,350,000 more than in the previous year for the issue ages under 35, thus less than 1 percent of the premium income when this has exceeded $300,000.
 

One could say that the company can figure the maximum rates for closing costs for every age, the total reserve of the new insurance policy would then amount to exactly zero, and the net premium of each individual issue age would then become equal to the usual net premium of the next higher age. If a company calculates its premiums from the start so that it every time adds a certain percentile loading, then the premiums for the higher issue ages would become very expensive [besides this for safety it would have to make the size of the commissions for the agents likewise dependable on the issue age which as already mentioned could lead to all sorts of difficulties]. The following table shows the various percentage rates which would remain for the bank from the loaded premium when the loaded premium is calculated in the usual method according to the Combined Experience Mortality Table and 3 1/2 percent interest and a loading of 12 1/2 percent. According to the usual method, the company would lose a ninth part or 11 1/9 percent of the premiums.
 
For Administrative Cost Stated as a Percent of the Loaded Premium, if the First Year Premium Is Reduced by: 
Issue Age Max Rate 1 % Insured Sum 1 1/4 % Insured Sum
20 
8.94% 
8.04% 
7.28% 
30 
8.46% 
8.55% 
7.91% 
40 
7.80% 
9.01% 
8.49% 
50 
7.14% 
9.41% 
8.99% 
60 
6.53% 
9.71% 
9.36% 



Additionally this table shows that the agency commissions which according to the earlier method were measured according to a determined percentage rate of the premium, according to the new method could be easily so determined that it becomes more advantageous for the company. After a significant commission in the first year after the closing of the insurance policy the agent will be readily satisfied with a smaller commission for the later years. If, for example, earlier a company paid as agents commission 6 percent of the premium, it would have only 5 1/9 percent for the other administrative costs. Now, if besides the closing commission it pays only 2 to 3 percent premium commission, there remains for it for most issue ages more than 5 1/9 percent.
 

As it concerns the reserve for an insurance policy that has existed for a number of years, the reserve would be somewhat less than according to the previous method. The difference between the two reserves will become smaller each year. As can easily be proved with the help of mathematical symbols, when the calculable rate of the closing commissions is equal to "a" percent of the insured amount, then the difference in the reserves is "a" percent of the insured sum lessened by "a" percent of the reserve calculated according to the old method.




Since the reserve gets bigger every year, the difference will become less with every year. For example, if the old reserve for an insurance policy of $100 has the value of:

The total reserve of an insurance bank for all insurance policies whose reserves are calculated according to the new principle has likewise a smaller value than if it were calculated according to the old method because each individual reserve is smaller. The difference will be relatively greater where the greatest number of insurance policies have not existed for very long. The difference here is also "a" percent of the insured sum lessened by "a" percent of the reserve calculated according to the old method when "a" percent of the insured sum was calculated as closing costs. For example, at the end of the year 1861, the Bank of Gotha had insurance inforce of $38,800,000 and a premium reserve of about $8,700,000. If the Bank of Gotha had figured from the beginning 1 percent closing costs, its reserves would be about: ($388,000 - $87,000) that is, $301,000 smaller.
 

Table I (attached) follows the reserve for an insurance policy of $100 for the issue ages of 30,40,and 50 through the whole term of the insurance policy. We see that the reserves with calculation of closing costs grow faster than the old reserves as it must be since from the second policy year on there is a higher, calculated net premium. The Tables II, III, and IV each follow a company of persons of the same age that insure themselves at the same time, and Table V gives a picture of the results achieved by an insurance bank when it reinsures every year such cohorts as are followed individually in Tables II, III, and IV. It shows that after a sequence of 60 years, with an insurance inforce of about $41,900,000, the old reserve would have grown to $12,644,900. The reserve with 1 percent closing costs would have grown to $12,352,000, and the reserve with 1 1/4 percent closing costs would have grown to $12,279,100. The difference between the old and the new reserve is, according to the above, with 1 percent closing costs $419,102 - $126,449 = $292,653 with 1 1/4 percent closing costs $419,102 - $126,449) * 5/4 = $365,817.
 

With the assumptions of Table V, a stationary state comes after 60 years, that is, as the consequence of annual cases of death, respectively, expirations of insurance policies, the same sum comes due which is annually newly insured. Therefore, in the future the insurance inforce, premium income, and reserve remain constant as long as the assumptions are met. In such a state, obviously, the net premiums and the calculable interest on those premiums and on the reserve capital must be as great as the annually payable insurance sum. From the existing insurance inforce after 60 years come:

$16,947,500 to members that join at age 30
$20,055,100 to members that join at age 40
$ 4,907,600 to members that join at age 50
 
 


If we now add to these values the expected premium and interest income sums and calculate premium incomes as follows:
 
 
 
 
Net Premium if the First Year's Premium is Reduced by
Issue Age
0 %
1 %
1 1/4 %
30
1.7973 
1.8490 
1.8620 
40
2.4841 
2.5428 
2.5575 
50
3.7083 
3.7792
3.7969 


ßx = Px + I / äx



We get the following income schedule which should cover the operations of the bank:
1) when the reserve is calculated without closing costs

Net Premium $1,021,661
3 1/2 percent interest thereon 35,758
3 1/2 percent interest on reserve 442,547
___________
Total $1,499,966
 
2) if the reserve is calculated with 1 percent closing costs
Net Premium (after deduction of $15,000) $1,031,547
3 1/2 percent interest thereon 36,104
3 1/2 percent interest on the reserve 432,327
-----------
Total $1,499,978
3) if the reserve is calculated with 1 1/4 percent closing costs
Net Premium (after deduction of $18,750) $1,034,019
3 1/2 percent interest thereon 36,190
3 1/2 percent interest on the reserve 429,768
------------
Total $1,499,977
The small differences of $34, $22, and $23 respectively against the required sum of $1,500,000 have their origin only in the round off that the numbers in Table V were subjected to (this is an alternative proof of the adequateness of the various reserves).

The above demonstrates clearly that a reserve calculated with a rate of closing costs that is not too high is completely sufficient for the solvency of the bank if the insurance polices exist until their normal end. If an insured allows his insurance to lapse while he is still alive, then this would not be awkward for the bank especially when it has used the precautionary rule that the negative reserves at the younger issue ages are not put up with the calculable amount but with the amount of zero, and when the agent does not receive a corresponding portion of the closing commissions or must pay out respectively in the case where the insurance lapses during the first year of its existence. On the contrary the collective reserves on such insurance policies go to the bank and flow to its profits. If at lapsing the insurance has been in existence long enough so that the insured can demand a buy back price, only the lower reserve can come into the calculations. As far as we know a bank is (not)obliged in such a case to pay the whole reserve, only a certain portion of the reserve. If a company, for example, customarily pays 3/4 of the reserve, then it can pay back just as much as when the reserve was calculated without closing costs as soon as the reserve has grown, for a $100 insured sum with 1 percent closing costs, to over $3 and, with 1 1/4 percent closing costs, to over $3 3/4 or, when the reserve without closing costs, had grown to $4 or $5, respectively, and it can pay back better since it has already paid itself for all closing costs.

II

Mr. Wilhelm Lazarus in the above mentioned paper "The Security of Life Insurance Institutions" (Rundschau der Versicherungen, 1857, p.317) called attention to it, noting that in the closing of the books of the Gresham Life Insurance Society on 12 November 1855, the liabilities of the society toward its insureds, even though it had been in existence for 7 years already, were valued lower than the value of premiums yet to be collected and yet a significant sum of the premiums to be used for the coverage of obligations had already been collected. A similar relationship shows itself in the balance sheet of the Great Britain published some time ago. After an almost 20 year existence, instead of a reserve, the Great Britain had as an asset the value of future premium income 529,469 pounds and as a liability the present value of the insured sum 494,707 pounds. Not only is there no reserve on hand, but the company figures for itself a net worth or profit of 34,762 pounds or $231,700 which sum is the present value of what the company will collect in excess of what it expects to pay at the close of


Note: 34,762 pounds is 231,700 talers. This is 6 2/3 talers per pound. The exchange rate to U.S. dollars at this time was 5 dollars per pound. Thus the exchange rate was 1 1/3 taler per dollar or $.75 for a taler.



 

the existing insurance policies. It is known that in the calculation of the net premium, the obligations of the insured are equalized with the obligations of the bank. If we continue on to the reserve calculation with net premium, then under special circumstances, as was shown in the first segment, the reserve of a portion of the insurance policies flowing out of the current year can become negative in calculation but this negative amount is of little significance as opposed to the reserves of all the insurance policies of that year and disappearing as opposed to the total reserves of all insurance policies from various years. It is then impossible, using calculations with net premiums, for the reserve to work out in the way it did with the Great Britain; and, in practice, a significant number of English companies calculate their reserves so that the premiums to be paid in the future by the insureds are calculated not according to the net premiums but according to a much higher premium and as a rule according to the loaded premium while the obligations of the company are valued according to the net premiums. As a consequence of this calculation, the reserve does not meet the demand that it be formed out of portion of the premium demanded for the risk after a one-time deduction of expenses covering calculable mortality and the addition of calculable interest. This method of calculating the reserves is clearly misleading. Apparently it is calculated according to the same principles as in the calculation with net premiums. One deducts from the value of the insured sum [the net single premium for insurance] the value of the premiums to be collected in the future, only instead of the net premium one takes the gross premiums. Herein lies the mistake. This principle for the calculation of reserves is generally only correct [even if it can be directly deduced] because and so far as it is a reformulation of the other principle according to which the reserve consists of the portion of the net premium not used up for mortality and the duly collected interest. If a company would enjoy a growing advantage out of closed insurance policy as soon as possible, then there is only the following way to sanction this procedure. So long as the loaded premium paid by an insured is higher than the net premium of that age into which the insured is gradually moving, the company figures as net premium the premium for one year term insurance, the excess of the loaded premium over the one year term premium it figures as profit. Once the insured has reached that age for which as issue age the net premium of the whole life insurance equals that premium which the insured pays, the whole life premium is charged. During the first (5 to 7) years when this forms absolutely no reserve, and only later would the company have a portion of the premium for reserve. Even with this model which borders on the imaginable, there is no talk about negative reserves. This is a new proof that this method of the English companies for calculating reserves with gross premiums is misleading.
 

The just described way of enjoying the advantage from an insurance policy as quickly as possible is surpassed by the method using gross premiums. Here in the moment of the closing of an insurance policy, a profit appears for as many percent of the total value of the future premium payments as the percentage loading included in the premiums, or since the value of the future premium payments measured according to net premiums is equal to the net single premium for the insurance there is a profit in the amount of just as much a percent of the net premium, as the loaded premium is higher than the net premium; for example, if a company figures according to the Combined Experience Table with an interest rate of 4 1/2 percent [in many cases, a higher interest rate is figured and the thus reduced net premiums are loaded on this much more] and 30 percent load of the net premiums, at the moment of the closing of the insurance policy the profit amounts to:
 
 
 
 
Issue Age 
Profit as Percent of Insured Sum 
20
6.63% 
30
8.16% 
37
9.60% 
40
10.34%
50
13.38% 
60
17.02%


(.30 ) Ax / 100


If such a company in a year closes insurance policies in the amount of $5,000,000 [on the average issue age 37] , in so doing it figures for itself a profit of not less than $480,000, that is,

 
$5,000,000 * 9.60%.


Therefore the first method cannot be sanctioned because nothing remains for the company to take care of business expenses if the net premium to be calculated for obligations of the bank [reserve and mortality] during the term of the contract reaches the magnitude of the loaded premium . If the company has significant stock capital, it can cover those costs with the interest on this capital when the business expansion is small. If the expansion of the business exceeds a certain limit then the interest on even the most significant capital will not be sufficient. One cannot sanction a model which, with the expansion of the business, does not offer growing means for the coverage of business costs -- with traditionally figuring banks, the means for the coverage of business costs and beyond that for the formation of profits, increase first in a determined relationship to premium income and then in a certain relationship to premium reserve in that the interest realized above the regular percentage rate [and this will probably not be higher than 3 1/2 percent in solid companies] flow not to the reserve but to the administrative accounts, respectively, the profit. This last income as a rule is completely missing in the irrationally figuring companies because usually they have a calculated interest rate that is so high that the actual interest on the reserve capital is actually much lower than that required by calculation.
 

The method of reserve calculation form the gross premiums suffers from the same problem, that it leaves nothing left for coverage of administrative costs from the premiums that will come in the future. It brings with it several disadvantages, that bring with them a soon and sure ruin of the company.
 

First we will compare the magnitude of the reserves that have been calculated according to the method of gross premiums with the magnitude of rationally calculated reserves. Table I shows in its fourth column the reserve for issue ages 30, 40, and 50 on the basis of a premium provided with an interest rate of 4 1/2 percent and a loading of 30 percent. The table shows that in the first 5 to 7 years, the reserve is negative. Later the reserve achieves a positive value and when the insurance policy continues until its end, the reserve finally grows to the full amount of the insurance sum. Table II shows in its seventh column the reserves which the bank puts up for the number of persons who at issue time had been 30 years old and insured altogether for $500,000 when the appropriate insurance policies are followed up to the natural end. Likewise Table III and IV show similar results for people who at issue age are 40 respectively 50 years old. Columns 8 and 9 in Tables II through IV also demonstrate the asset and liability postings that are so beloved in the English accounting. The difference of the numbers from columns 8 and 9 give the reserves under column 7. Table V contains the summary of the three preceding tables. When we add numbers of the individual columns from the top down, we get those results that an insurance company achieves when it insures such groups of insurance policies each year. The row sums combined out of the 3 tables gives then the numbers for Table V. The premium rates [Column 3] that were the basis for the premium income are the premiums of 4 1/2 percent raised by 30 percent. The columns 8 and 9 were omitted but they could be developed by the addition of the 3 Tables II to IV.

Table V shows that the total reserve becomes positive only in the 13th year, and that finally after 60 years, where under the assumed circumstances, likewise takes place a stationary condition, the reserve is about $4,000,000 smaller than the reserve found by rational calculation in columns 4, 5, and 6. This difference amounts to about 10 percent of the insurance sum, more than 30 percent of the rational reserve, and about 350 percent of the premium income. These relationships become even more colossal if we presume a steadily growing business as it happens in Table IV. Here after 30 years the difference between the reserves from the gross premium and the rationally calculated reserve already amounts to about $9,000,000, that is, more than 50 percent of the rational reserve.
 

To examine further what influence the calculation with gross premiums has on the development of the business, we will first assume that no insurance policy expires before its regular end. If the company does fora longer time equal business then, and this is certainly an unnatural phenomenon, the profit lessens from year to year. If, for example, we use the numbers of Table V and assume the existence of a fund from which the company can withdraw the artificially calculated profit for as long as the actually collected premium moneys do not reach so far and if we further assume that all funds draw interest at 4 1/2 percent, and if we calculate agency commissions as 1/2 of the first annual premium and only 1 percent of all later premiums and as administrative costs annually $20,000 multiplied by .1 percent of the insurance sum, the company will achieve profits as they are put together in the following table:
 
Business Year
Profit
Premium Income
Profit as a Percent of Premium Income
1
$120,000 
$43,800
274%
5
$111,400
$213,900
52%
10
$101,700
$413,800
25%
15
$92,200
$597,300
15%
20
$83,400
$760,700
11%
30
$67,600
$1,012,400
7%
60
$50,500
$1,209,500
4%

(table from page 28)

Profit therefore decreases by about 58 percent in the course of time since with an equal or even business, the profit becomes less each year. It is a necessary consequence that the whole profit that a policy can bring with regular existence will be used up in the year of the end of the insurance policy. If the business continues in the same way, there would result year after year the same profit if the business expense remains the same. But these business expenses grow with the expansion of the business; therefore the profit must decline. Conversely, if a company wants to achieve equal profits in each following year, it must do more and more business so that the profit will not sink under a certain percent of the premium income [In the last case, the profit from a certain point in time on would have to grow proportionately to the premium income].
 

This appearance which characterizes the whole system as unhealthy emerges even without lapses. But what happens when a large part of the insurance policies lapse time and again? Then here emerges the main problem of this system. The lapsing of insurance policies occurs, to a large extent, in insurance policies that are not very old yet, where as a rule the reserve is still negative. With the lapsing of such an insurance policy, the negative reserve, respectively, the excessive asset posting disappear; the company now must cover this disappearance by new insurance policies which bring with them negative reserves and only with growth beyond that comes the attainment of the annual profit. Since with growing expansion, the number of lapsing insurance policies also grows then the business has to grow faster and faster. Since the accomplishment of this demand requires superhuman efforts, the sad consequences of this unhealthy system are very soon demonstrated here too. Even with the calculation with gross premiums according to which the reserve, the debt of the bank to its insureds, is established so small in the balance sheet even sometimes conversely as a debt of the insured, the companies very soon get into difficulties because the expansion is not increasing sufficiently [the losses in interest income because of the too high assumed interest rate are not even calculated here]. To figure out the proper profit, the to cover the no longer manageable deficit, the companies are forced to extrapolate all kinds of doubtful postings in accruals, and it is very difficult for the public to judge their value. When this manuever finally will not work any more, one attempts to get rid of the whole business by ceding it to another company. It would hardly seem possible that a company would get rid of a business in such a sickly state; but the experience shows that every year in England so and so many amalgamations and business transfers occur. How a company can feel motivated to take over such an unhealthy business cannot even be conceived in a German imagination. Such companies can only speculate that in the next years the premium income will exceed the expenditures for mortality and that they could in some one form calculate the excess part of the premium income as gain. When it finally will not go any farther, the business collapses; it does no harm when the many insureds remain unsatisfied. The executives of the business and their shareholders, these too have enjoyed their benefits. Nothing keeps them from establishing a new business under some other name. A fraternal burial society which is arranged as badly as possible, which collects contributions according to the principle of the present need, is to be preferred to such a life insurance company. When the burial society cannot continue on the whole, the members have paid as little as possible, and certainly too little; however, the policy holders of such a company have paid sufficient amounts and yet come out empty in the end.
 

When an insured, who has already been insured long enough so that for him, individual calculations result in a positive reserve, wants to give up his insurance policy, the company according to calculations could concede a buy back price, but in reality there is no money, for against the positive reserve of the long term insurance falls the negative reserve of a younger insured, and both reserves together cancel each other partly or fully. Therefore, as a rule, one seeks vainly in the insurance terms of such companies for a rule which obliges the company to buy back the policy; many companies declare frequently, naturally without adding the true reason, that a buy back in mass would be disadvantageous to the company. Naturally when one has to pay more than one has, that can hurt.
 

We can establish different consequences of the unhealthy system of calculating with gross premiums. We want to mention only the following. When a company suddenly ceases every entry of new insurance policies, in the rational calculation the reserve will grow for a time if a stationary state had not already been entered. In any case the reserve grows less than when the new insurance policies are added. It is reversed in a system of gross premiums. If insurance policies are not being added any more during a series of years, the reserve will grow stronger than when new insurance policies are added.
 

Several companies, that evidently are calculating with gross premiums, declare that they in the calculations of the value of future premium payments set apart a sufficient portion of the loaded premiums for the administrative costs. Suppose now that the value of the insured sums is calculated not only according to the same mortality table, but also the same interest rate as the value of the future premium payments [a higher interest rate in the calculation of the value of the insured sums, and many companies calculate this way, obviously would form this value as less and thereby cancel the setting aside of a part of the loaded premium for the administrative costs when not leading to worse results]; the partitioning of the whole loading would lead to a rational establishment of the reserve. As a rule, the premiums have colossal loadings, occasionally 40% or more of the net premiums and above; if now about 5 percent of it is partitioned, then the premiums that are to be used for the calculation of future premium income are loaded so sufficiently that the reserve for the individual insurance policy here also needs several years to go over from the negative to the positive. If the annual business profit with equal additions and without losses by lapsing would become equal [the same year after year], that is, in the sum, not in the percent of premium income then since a portion of the insurance policies will lapse to achieve the equal gain, the annual additions must grow or increase, and since with growing additions there also grows the number of lapsing insurance policies, the additions must grow for so much more. Should the annual profit not sink below a certain percentage rate of the premium income, the annual additions must naturally grow so much faster. Here as above, the condition of the company is tied to the demand that the annual additions must grow faster each year. Also for this case, there result the same consequences as when the reserves are calculated according to the loaded premiums only here, they occur somewhat slower and milder.
 

If we summarize the above, then the mistakes of the reserve calculation from gross premiums which are higher than the net premiums calculated for the risk, but in general equal to the loaded premiums, consists of the following:

 
1) At the close of an insurance policy, there is calculated an artificial gain which consequently may not be realized and in so and so many cases is not actually realized;
 

2) The reserve of the individual insurance policy is in and for itself already too low besides that the negative reserves of the younger insurance policies partly or wholly cancel the positive reserves of the older insurance policies;
 

3) The gain mentioned under 1) is the sole or main source

a) for absorbing the administrative costs,
b) for the coverage of the disappearing negative reserves as consequence of the premature expiration of the insurance policies,
c) for the annual profit.
Since both the first postings [ a) and b)] are necessary expenses and in addition growing ones, but since the profit mentioned under 1) cannot always grow further or what amounts to the same thing since the annual addition or growth cannot go higher forever with each following year, then the annual profit mentioned under c) sooner or later must let up and finally a deficit will come into existence and the bank will become insolvent.



Reserves for an endowment at age 90 with a face amount of $100 and premiums payable annually, calculated according to the Combined Experience Table.
 
 

                      Table I.
                      Zillmer Reserve Calculations
                      Endowment @90
                                  Interest Rate 3.5%
             Issue Age 30
             Net Premium       1.797252
         I= 0                I=       1     I=       1.25
    ----------------| ----------------| ----------------|
     Initial Terminal Initial  Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves Reserves Reserves
    ------------------------------------------------------
 1    1.7973   1.0263   0.7973   0.0366   0.5473  -0.2109
 2    2.8236   2.0824   1.8338   1.1033   1.5864   0.8585
 3    3.8797   3.1685   2.9005   2.2002   2.6557   1.9581
 4    4.9658   4.2859   3.9974   3.3287   3.7554   3.0894
 5    6.0831   5.4359   5.1260   4.4903   4.8867   4.2539
 6    7.2332   6.6190   6.2875   5.6852   6.0511   5.4518
 7    8.4163   7.8367   7.4825   6.9151   7.2490   6.6847
 8    9.6340   9.0905   8.7123   8.1814   8.4819   7.9541
 9   10.8878  10.3811   9.9787   9.4849   9.7514   9.2608
10   12.1783  11.7101  11.2821  10.8272  11.0581  10.6065
11   13.5073  13.0794  12.6244  12.2102  12.4037  11.9929
12   14.8767  14.4899  14.0075  13.6348  13.7902  13.4211
13   16.2872  15.9415  15.4321  15.1009  15.2183  14.8907
14   17.7387  17.4306  16.8981  16.6049  16.6880  16.3985
15   19.2279  18.9528  18.4022  18.1423  18.1958  17.9397
16   20.7501  20.5055  19.9396  19.7106  19.7370  19.5118
17   22.3028  22.0830  21.5078  21.3038  21.3091  21.1090
18   23.8802  23.6846  23.1011  22.9214  22.9063  22.7307
19   25.4818  25.3086  24.7187  24.5617  24.5279  24.3750
20   27.1059  26.9545  26.3590  26.2240  26.1723  26.0414
21   28.7517  28.6203  28.0213  27.9065  27.8387  27.7281
22   30.4176  30.3045  29.7038  29.6075  29.5254  29.4333
23   32.1017  32.0050  31.4048  31.3250  31.2305  31.1550
24   33.8022  33.7198  33.1223  33.0570  32.9523  32.8913
25   35.5171  35.4489  34.8543  34.8034  34.6886  34.6420
26   37.2462  37.1891  36.6007  36.5609  36.4393  36.4039
27   38.9863  38.9387  38.3582  38.3281  38.2012  38.1754
28   40.7360  40.6982  40.1253  40.1052  39.9727  39.9569
29   42.4954  42.4647  41.9024  41.8893  41.7542  41.7455
30   44.2619  44.2360  43.6866  43.6783  43.5427  43.5389

             Issue Age 30 (Continued)

     I= 0                I=       1     I=       1.25
    ----------------| ----------------| ----------------|
     Initial Terminal Initial  Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves Reserves Reserves
    -----------------------------------------------------
31   46.0332  46.0064  45.4756  45.4665  45.3362  45.3315
32   47.8037  47.7736  47.2637  47.2513  47.1287  47.1208
33   49.5709  49.5334  49.0486  49.0288  48.9180  48.9026
34   51.3307  51.2838  50.8260  50.7967  50.6998  50.6749
35   53.0811  53.0210  52.5939  52.5512  52.4721  52.4338
36   54.8182  54.7418  54.3485  54.2892  54.2310  54.1761
37   56.5391  56.4441  56.0865  56.0085  55.9733  55.8996
38   58.2413  58.1243  57.8058  57.7055  57.6969  57.6008
39   59.9215  59.7814  59.5028  59.3792  59.3981  59.2787
40   61.5787  61.4155  61.1765  61.0296  61.0759  60.9332
41   63.2127  63.0242  62.8269  62.6545  62.7304  62.5620
42   64.8215  64.6072  64.4517  64.2532  64.3593  64.1647
43   66.4044  66.1636  66.0505  65.8252  65.9620  65.7406
44   67.9608  67.6942  67.6225  67.3711  67.5379  67.2903
45   69.4914  69.1987  69.1684  68.8907  69.0876  68.8137
46   70.9959  70.6789  70.6879  70.3857  70.6109  70.3124
47   72.4761  72.1380  72.1829  71.8594  72.1096  71.7897
48   73.9353  73.5777  73.6566  73.3135  73.5870  73.2474
49   75.3750  75.0023  75.1107  74.7523  75.0447  74.6898
50   76.7995  76.4206  76.5495  76.1848  76.4870  76.1259
51   78.2179  77.8448  77.9821  77.6232  77.9232  77.5678
52   79.6420  79.2939  79.4205  79.0868  79.3651  79.0350
53   81.0911  80.7952  80.8841  80.6032  80.8323  80.5551
54   82.5925  82.3844  82.4004  82.2082  82.3524  82.1642
55   84.1816  84.1150  84.0055  83.9561  83.9615  83.9164
56   85.9122  86.0601  85.7534  85.9207  85.7136  85.8859
57   87.8574  88.3378  87.7180  88.2212  87.6831  88.1920
58   90.1350  91.1447  90.0184  91.0562  89.9893  91.0340
59   92.9420  94.8211  92.8534  94.7693  92.8313  94.7564
60   96.6184   0.0000  96.5666   0.0000  96.5536   0.0000


    Table I.
    Zillmer Reserve Calculations
    Endowment @90, Load 30% NSP
    Interest Rate 4.5%
             Issue Age 30
             Net Premium       1.609444
         I= 0                I=  8.1619
    ----------------| ----------------|
     Initial Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves
    ------------------------------------
 1    1.6094   0.8465  -6.5525  -7.2463
 2    2.4560   1.7234  -5.6369  -6.2978
 3    3.3329   2.6312  -4.6884  -5.3160
 4    4.2406   3.5714  -3.7065  -4.2990
 5    5.1808   4.5458  -2.6896  -3.2451
 6    6.1552   5.5550  -1.6356  -2.1535
 7    7.1645   6.6010  -0.5440  -1.0221
 8    8.2104   7.6857   0.5873   0.1511
 9    9.2951   8.8101   1.7605   1.3672
10   10.4195   9.9763   2.9767   2.6287
11   11.5858  11.1869   4.2381   3.9380
12   12.7963  12.4430   5.5475   5.2967
13   14.0525  13.7451   6.9061   6.7051
14   15.3546  15.0902   8.3145   8.1600
15   16.6997  16.4741   9.7694   9.6568
16   18.0836  17.8947  11.2663  11.1933
17   19.5041  19.3463  12.8027  12.7634
18   20.9557  20.8287  14.3728  14.3668
19   22.4381  22.3404  15.9762  16.0019
20   23.9499  23.8812  17.6114  17.6685
21   25.4906  25.4495  19.2779  19.3648
22   27.0589  27.0437  20.9742  21.0891
23   28.6532  28.6623  22.6986  22.8398
24   30.2717  30.3032  24.4492  24.6147
25   31.9127  31.9668  26.2241  26.4140
26   33.5762  33.6497  28.0234  28.2343
27   35.2592  35.3507  29.8437  30.0741
28   36.9602  37.0703  31.6836  31.9341
29   38.6798  38.8057  33.5435  33.8111
30   40.4152  40.5547  35.4206  35.7029
31   42.1642  42.3115  37.3123  37.6030
32   43.9210  44.0735  39.2125  39.5089
33   45.6830  45.8365  41.1183  41.4157
34   47.4459  47.5981  43.0252  43.3211
35   49.2076  49.3543  44.9306  45.2206
36   50.9637  51.1015  46.8301  47.1105
37   52.7110  52.8374  48.7199  48.9880
38   54.4468  54.5578  50.5975  50.8489
39   56.1673  56.2617  52.4583  52.6918
40   57.8711  57.9486  54.3012  54.5164

    Endowment @90, Load 30% NSP
    Interest Rate 4.5%
             Issue Age 30 (Continued)
             Net Premium       1.609444
         I= 0                I=  8.1619
    ----------------| ----------------|
     Initial Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves
    ------------------------------------

41   59.5580  59.6158  56.1258  56.3197
42   61.2253  61.2627  57.9292  58.1010
43   62.8721  62.8881  59.7104  59.8591
44   64.4976  64.4925  61.4685  61.5944
45   66.1019  66.0753  63.2038  63.3064
46   67.6847  67.6380  64.9158  64.9967
47   69.2474  69.1840  66.6061  66.6688
48   70.7935  70.7148  68.2783  68.3245
49   72.3242  72.2346  69.9340  69.9684
50   73.8440  73.7533  71.5779  71.6110
51   75.3627  75.2837  73.2205  73.2664
52   76.8931  76.8472  74.8758  74.9575
53   78.4566  78.4743  76.5669  76.7174
54   80.0837  80.2054  78.3268  78.5898
55   81.8148  82.1016  80.1992  80.6407
56   83.7110  84.2472  82.2502  82.9614
57   85.8566  86.7787  84.5709  85.6996
58   88.3881  89.9251  87.3090  89.1028
59   91.5346  94.0843  90.7122  93.6015
60   95.6938   0.0000  95.2109   0.0000


                      Table I.
                      Zillmer Reserve Calculations
                      Endowment @90
                                  Interest Rate 3.5%
    Issue Age 40
    Net Premium        2.48414
       I =       0       I =          1   I =        1.25
    ----------------| ----------------| ----------------|
    Initial  Terminal Initial  Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves Reserves Reserves
    ------------------------------------------------------
 1    1.7973   1.5510   0.7973   0.5665   0.5473   0.3203
 2    3.3482   3.1486   2.3637   2.1800   2.1176   1.9379
 3    4.9458   4.7926   3.9773   3.8405   3.7352   3.6025
 4    6.5899   6.4793   5.6378   5.5441   5.3998   5.3103
 5    8.2765   8.2033   7.3413   7.2854   7.1075   7.0559
 6   10.0006   9.9620   9.0826   9.0616   8.8531   8.8365
 7   11.7593  11.7487  10.8589  10.8662  10.6338  10.6456
 8   13.5460  13.5627  12.6634  12.6984  12.4428  12.4823
 9   15.3600  15.4022  14.4956  14.5562  14.2795  14.3447
10   17.1994  17.2663  16.3535  16.4390  16.1420  16.2321
11   19.0635  19.1531  18.2362  18.3446  18.0294  18.1425
12   20.9504  21.0606  20.1419  20.2712  19.9398  20.0739
13   22.8579  22.9866  22.0685  22.2165  21.8711  22.0240
14   24.7839  24.9290  24.0138  24.1782  23.8212  23.9906
15   26.7262  26.8874  25.9755  26.1563  25.7878  25.9735
16   28.6846  28.8583  27.9535  28.1469  27.7707  27.9690
17   30.6556  30.8400  29.9441  30.1484  29.7663  29.9755
18   32.6373  32.8329  31.9457  32.1612  31.7728  31.9933
19   34.6301  34.8336  33.9584  34.1820  33.7905  34.0191
20   36.6309  36.8399  35.9792  36.2083  35.8163  36.0504
21   38.6371  38.8451  38.0055  38.2336  37.8476  38.0807
22   40.6424  40.8467  40.0308  40.2552  39.8780  40.1073
23   42.6440  42.8399  42.0524  42.2683  41.9045  42.1254
24   44.6372  44.8225  44.0656  44.2707  43.9227  44.1328
25   46.6198  46.7901  46.0680  46.2580  45.9301  46.1249
26   48.5873  48.7391  48.0552  48.2265  47.9222  48.0984
27   50.5364  50.6672  50.0238  50.1738  49.8956  50.0505
28   52.4644  52.5702  51.9711  52.0959  51.8478  51.9773
29   54.3674  54.4471  53.8932  53.9916  53.7746  53.8777
30   56.2444  56.2979  55.7889  55.8609  55.6750  55.7517
31   58.0952  58.1201  57.6582  57.7013  57.5489  57.5966
32   59.9173  59.9129  59.4985  59.5121  59.3938  59.4118
33   61.7102  61.6758  61.3093  61.2926  61.2091  61.1968
34   63.4731  63.4094  63.0898  63.0435  62.9940  62.9520
35   65.2066  65.1134  64.8407  64.7646  64.7492  64.6774
36   66.9107  66.7899  66.5618  66.4578  66.4746  66.3748
37   68.5872  68.4426  68.2551  68.1270  68.1721  68.0481
38   70.2399  70.0733  69.9243  69.7740  69.8454  69.6992
39   71.8705  71.6867  71.5713  71.4036  71.4964  71.3328
40   73.4840  73.2933  73.2009  73.0262  73.1301  72.9594

    Issue Age 40 (Continued)

   I =       0       I =          1   I =        1.25
    ----------------| ----------------| ----------------|
    Initial  Terminal Initial  Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves Reserves Reserves
    ------------------------------------------------------

41   75.0905  74.9063  74.8234  74.6554  74.7567  74.5926
42   76.7035  76.5476  76.4526  76.3130  76.3899  76.2544
43   78.3448  78.2480  78.1103  78.0305  78.0517  77.9761
44   80.0453  80.0480  79.8278  79.8485  79.7734  79.7986
45   81.8453  82.0081  81.6457  81.8282  81.5959  81.7832
46   83.8053  84.2113  83.6254  84.0534  83.5804  84.0139
47   86.0085  86.7910  85.8506  86.6589  85.8112  86.6259
48   88.5883  89.9702  88.4562  89.8699  88.4231  89.8448
49   91.7675  94.1342  91.6672  94.0756  91.6421  94.0609
50   95.9315   0.0000  95.8728   0.0000  95.8581   0.0000



    Table I.
    Zillmer Reserve Calculations
    Endowment @90, Load 30% NSP
    Interest Rate 4.5%
    Issue Age 40
    Net Premium       2.265013
        I=          0 I=        10.3406
    ----------------| ----------------|
    Initial  Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves
    ------------------------------------
 1    2.2650   1.3447  -8.0756  -8.8569
 2    3.6097   2.7400  -6.5919  -7.3172
 3    5.0050   4.1864  -5.0522  -5.7213
 4    6.4514   5.6806  -3.4563  -4.0726
 5    7.9456   7.2179  -1.8076  -2.3764
 6    9.4829   8.7958  -0.1114  -0.6352
 7   11.0608  10.4083   1.6298   1.1440
 8   12.6733  12.0550   3.4090   2.9609
 9   14.3200  13.7343   5.2259   4.8139
10   15.9993  15.4458   7.0789   6.7024
11   17.7108  17.1879   8.9674   8.6246
12   19.4529  18.9588  10.8896  10.5787
13   21.2238  20.7567  12.8437  12.5625
14   23.0217  22.5795  14.8275  14.5738
15   24.8445  24.4274  16.8388  16.6127
16   26.6924  26.2969  18.8778  18.6755
17   28.5619  28.1864  20.9405  20.7604
18   30.4514  30.0965  23.0254  22.8681
19   32.3615  32.0242  25.1331  24.9951
20   34.2892  33.9671  27.2601  27.1389
21   36.2321  35.9185  29.4039  29.2921
22   38.1835  37.8758  31.5571  31.4518
23   40.1408  39.8341  33.7168  33.6126
24   42.0991  41.7910  35.8776  35.7718
25   44.0560  43.7417  38.0368  37.9243
26   46.0068  45.6827  40.1893  40.0659
27   47.9477  47.6109  42.3309  42.1935
28   49.8759  49.5220  44.4585  44.3023
29   51.7870  51.4146  46.5673  46.3906
30   53.6796  53.2885  48.6556  48.4582
31   55.5535  55.1405  50.7232  50.5018
32   57.4055  56.9699  52.7668  52.5203
33   59.2349  58.7754  54.7853  54.5126
34   61.0404  60.5576  56.7776  56.4790
35   62.8226  62.3158  58.7440  58.4190
36   64.5808  64.0517  60.6840  60.3344
37   66.3167  65.7690  62.5994  62.2293
38   68.0340  67.4694  64.4943  64.1055
39   69.7344  69.1577  66.3705  65.9684
40   71.4227  70.8446  68.2334  67.8298

    Table I.
    Zillmer Reserve Calculations
    Endowment @90, Load 30% NSP
    Interest Rate 4.5%
    Issue Age 40 (Continued)
    Net Premium       2.265013
        I=          0 I=        10.3406
    ----------------| ----------------|
    Initial  Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves
    ------------------------------------

41   73.1096  72.5447  70.0948  69.7056
42   74.8097  74.2814  71.9706  71.6219
43   76.5464  76.0888  73.8869  73.6163
44   78.3538  78.0118  75.8813  75.7381
45   80.2768  80.1181  78.0031  78.0622
46   82.3831  82.5015  80.3272  80.6920
47   84.7665  85.3135  82.9570  83.7948
48   87.5785  88.8086  86.0598  87.6514
49   91.0736  93.4288  89.9164  92.7493
50   95.6938   0.0000  95.0143   0.0000






                      Table I.
                      Zillmer Reserve Calculations
                      Endowment @90
                                  Interest Rate 3.5%
    Issue Age 50
    Net Premium       3.708312
       I =       0       I =          1   I =        1.25
    ----------------| ----------------| ----------------|
    Initial  Terminal Initial  Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves Reserves Reserves
    ------------------------------------------------------
 1    3.7083   2.2806   2.7083   1.2806   2.4583   1.0591
 2    5.9889   4.5862   5.0117   3.6098   4.7674   3.3935
 3    8.2945   6.9142   7.3404   5.9616   7.1018   5.7506
 4   10.6225   9.2618   9.6916   8.3333   9.4589   8.1276
 5   12.9701  11.6290  12.0628  10.7247  11.8359  10.5244
 6   15.3373  14.0112  14.4536  13.1313  14.2327  12.9364
 7   17.7195  16.4065  16.8597  15.5511  16.6447  15.3616
 8   20.1148  18.8153  19.2789  17.9845  19.0699  17.8005
 9   22.5236  21.2336  21.7117  20.4276  21.5088  20.2490
10   24.9419  23.6585  24.1543  22.8773  23.9573  22.7042
11   27.3668  26.0823  26.6034  25.3259  26.4126  25.1583
12   29.7906  28.5016  29.0514  27.7699  28.8666  27.6079
13   32.2099  30.9108  31.4949  30.2038  31.3162  30.0472
14   34.6191  33.3071  33.9282  32.6246  33.7555  32.4735
15   37.0155  35.6853  36.3485  35.0272  36.1818  34.8814
16   39.3936  38.0411  38.7505  37.4071  38.5897  37.2667
17   41.7495  40.3716  41.1299  39.7614  40.9750  39.6262
18   44.0799  42.6717  43.4836  42.0851  43.3345  41.9551
19   46.3800  44.9404  45.8068  44.3769  45.6634  44.2521
20   48.6487  47.1774  48.0981  46.6369  47.9605  46.5172
21   50.8858  49.3798  50.3575  48.8618  50.2255  48.7471
22   53.0882  51.5469  52.5820  51.0510  52.4554  50.9412
23   55.2552  53.6777  54.7706  53.2036  54.6495  53.0986
24   57.3860  55.7730  56.9227  55.3204  56.8069  55.2202
25   59.4813  57.8327  59.0391  57.4012  58.9285  57.3056
26   61.5410  59.8591  61.1194  59.4483  61.0139  59.3573
27   63.5674  61.8567  63.1660  61.4663  63.0657  61.3799
28   65.5650  63.8277  65.1835  63.4575  65.0882  63.3755
29   67.5360  65.7779  67.1742  65.4276  67.0838  65.3501
30   69.4862  67.7196  69.1439  67.3893  69.0584  67.3161
31   71.4280  69.6693  71.1052  69.3589  71.0245  69.2902
32   73.3776  71.6531  73.0743  71.3630  72.9985  71.2988
33   75.3614  73.7085  75.0780  73.4394  75.0071  73.3798
34   77.4168  75.8841  77.1539  75.6373  77.0881  75.5826
35   79.5924  78.2532  79.3512  78.0307  79.2909  77.9814
36   81.9615  80.9162  81.7441  80.7209  81.6897  80.6777
37   84.6245  84.0343  84.4337  83.8709  84.3860  83.8348
38   87.7426  87.8770  87.5830  87.7530  87.5431  87.7255
39   91.5853  92.9100  91.4641  92.8375  91.4338  92.8214
40   96.6184   0.0000  96.5475   0.0000  96.5297   0.0000

    Table I.
    Zillmer Reserve Calculations
    Endowment @90, Load 30% NSP
    Interest Rate 4.5%
    Issue Age 50
    Net Premium         3.4654
    I=              0 I=        13.3771
    ----------------| ----------------|
    Initial  Terminal Initial  Terminal
    Reserves Reserves Reserves Reserves
    ------------------------------------
 1    3.4654   2.0603  -9.9117 -11.0412
 2    5.5257   4.1548  -7.5758  -8.6666
 3    7.6202   6.2811  -5.2012  -6.2558
 4    9.7465   8.4369  -2.7904  -3.8116
 5   11.9023  10.6223  -0.3462  -1.3338
 6   14.0877  12.8333   2.1316   1.1729
 7   16.2987  15.0680   4.6383   3.7065
 8   18.5334  17.3270   7.1719   6.2678
 9   20.7924  19.6069   9.7332   8.8526
10   23.0723  21.9046  12.3180  11.4577
11   25.3700  24.2126  14.9231  14.0744
12   27.6780  26.5274  17.5398  16.6989
13   29.9928  28.8434  20.1643  19.3248
14   32.3088  31.1578  22.7902  21.9487
15   34.6232  33.4649  25.4141  24.5644
16   36.9303  35.7603  28.0298  27.1669
17   39.2257  38.0408  30.6323  29.7524
18   41.5062  40.3010  33.2178  32.3150
19   43.7664  42.5394  35.7804  34.8528
20   46.0048  44.7555  38.3182  37.3654
21   48.2209  46.9459  40.8308  39.8488
22   50.4113  49.1094  43.3142  42.3017
23   52.5748  51.2448  45.7671  44.7228
24   54.7102  53.3525  48.1882  47.1124
25   56.8179  55.4319  50.5778  49.4699
26   58.8973  57.4849  52.9353  51.7976
27   60.9503  59.5159  55.2630  54.1003
28   62.9813  61.5269  57.5657  56.3803
29   64.9923  63.5236  59.8457  58.6441
30   66.9890  65.5187  62.1095  60.9061
31   68.9841  67.5293  64.3715  63.1857
32   70.9947  69.5833  66.6511  65.5144
33   73.0487  71.7209  68.9798  67.9380
34   75.1863  73.9951  71.4034  70.5164
35   77.4605  76.4862  73.9818  73.3407
36   79.9516  79.3049  76.8061  76.5365
37   82.7703  82.6307  80.0019  80.3072
38   86.0961  86.7643  83.7726  84.9937
39   90.2297  92.2284  88.4591  91.1888
40   95.6938   0.0000  94.6542   0.0000

        Table II
Overview of the business outcomes which result when a life
insurance bank insures a cohort of 30 year olds, each one for the
same sum, in a total of $500,000.  The annual premium is
percent.

                                     Life Reserves on the Basis
  Business Insurance         Premium  3 1/2 % Interest &
  Year     Inforce  Claims   Income   I = 0%   I = 1%  I=11/4%
  - --- ---- --- ---- --- ---- --- ---- --- ---- --- ---- --- ---
       0   500000
        1   495800     4200    10458     5100      181    -1045
        2   491500     4300    10370    10200     5422     4219
        3   487200     4300    10281    15400    10719     9540
        4   482900     4300    10191    20700    16074    14919
        5   478500     4400    10101    26000    21486    20355
        6   474100     4400    10009    31400    26954    25847
        7   469600     4500     9917    36800    32473    31391
        8   465100     4500     9822    42300    38052    36995
        9   460500     4600     9728    47800    43678    42646
       10   455800     4700     9632    53400    49350    48344
       11   451100     4700     9534    59000    55080    54100
       12   446300     4800     9436    64700    60852    59898
       13   441400     4900     9335    70400    66655    65728
       14   436400     5000     9233    76100    72464    71563
       15   431300     5100     9128    81700    78248    77374
       16   426000     5300     9021    87400    83967    83120
       17   420500     5500     8910    92900    89583    88763
       18   414800     5700     8795    98200    95078    94287
       19   408900     5900     8676   103500   100433    99669
       20   402700     6200     8553   108500   105604   104869
       21   396300     6400     8423   113400   110594   109886
       22   389600     6700     8289   118100   115351   114672
       23   382600     7000     8149   122500   119850   119199
       24   375300     7300     8003   126600   124063   123441
       25   367700     7600     7850   130300   127972   127379
       26   359700     8000     7691   133800   131510   130945
       27   351400     8300     7524   136800   134685   134149
       28   342700     8700     7350   139500   137440   136932
       29   333700     9000     7168   141700   139785   139305
       30   324300     9400     6980   143500   141649   141197
       31   314500     9800     6783   144700   142992   142568
       32   304200    10300     6578   145300   143739   143341
       33   293500    10700     6363   145400   143899   143529
       34   282400    11100     6139   144800   143450   143106
       35   270900    11500     5907   143600   142361   142043
       36   259000    11900     5666   141800   140609   140316
       37   246700    12300     5417   139200   138173   137904
       38   234000    12700     5160   136000   135031   134786
       39   221000    13000     4894   132100   131228   131006
       40   207700    13300     4623   127600   126759   126558
        Table II (Continued)

                                     Life Reserves on the Basis
  Business Insurance         Premium  3 1/2 % Interest &
  Year     Inforce  Claims   Income   I = 0%   I = 1%  I=11/4%
  - --- ---- --- ---- --- ---- --- ---- --- ---- --- ---- --- ---
       41   194200    13500     4344   122400   121675   121495
       42   180600    13600     4062   116700   116041   115882
       43   166900    13700     3778   110400   109862   109721
       44   153200    13700     3491   103700   103213   103089
       45   139600    13600     3204    96600    96171    96064
       46   126300    13300     2920    89300    88897    88805
       47   113300    13000     2642    81700    81417    81338
       48   100700    12600     2370    74100    73827    73760
       49    88600    12100     2106    66500    66231    66175
       50    77100    11500     1853    58900    58739    58693
       51    66300    10800     1613    51600    51464    51427
       52    56300    10000     1387    44600    44526    44497
       53    47100     9200     1178    38100    37964    37941
       54    38800     8300      985    32000    31897    31880
       55    31400     7400      812    26400    26362    26350
       56    25000     6400      657    21500    21480    21471
       57    19400     5600      523    17100    17115    17109
       58    14700     4700      406    13400    13385    13382
       59    10800     3900      307    10200    10235    10234
       60        0    10800      226        0        0        0

        Table II (Continued)


           Age 30                     Reserves Asset    Liability
 Business Insurance         Premium  4 1/2%   NPV      NPV
  Year     Inforce  Claims   Income   30% Load Premiums Benefits
       0   500000
       1   495800     4200    10458   -35900   173800   137900
       2   491500     4300    10370   -31000   170800   139900
       3   487200     4300    10281   -25900   167700   141900
       4   482900     4300    10191   -20800   164600   143900
       5   478500     4400    10101   -15500   161500   146000
       6   474100     4400    10009   -10200   158300   148200
       7   469600     4500     9917    -4800   155100   150300
       8   465100     4500     9822      700   151800   152600
       9   460500     4600     9728     6300   148500   154800
      10   455800     4700     9632    12000   145100   157100
      11   451100     4700     9534    17800   141700   159500
      12   446300     4800     9436    23600   138200   161800
      13   441400     4900     9335    29600   134600   164300
      14   436400     5000     9233    35600   131000   166700
      15   431300     5100     9128    41600   127400   169100
      16   426000     5300     9021    47700   123700   171400
      17   420500     5500     8910    53700   119900   173600
      18   414800     5700     8795    59600   116100   175700
      19   408900     5900     8676    65400   112300   177700
      20   402700     6200     8553    71200   108400   179600
      21   396300     6400     8423    76700   104500   181200
      22   389600     6700     8289    82200   100500   182700
      23   382600     7000     8149    87400    96500   183900
      24   375300     7300     8003    92400    92500   184900
      25   367700     7600     7850    97100    88500   185600
      26   359700     8000     7691   101600    84400   186000
      27   351400     8300     7524   105700    80300   186000
      28   342700     8700     7350   109400    76300   185700
      29   333700     9000     7168   112800    72200   185100
      30   324300     9400     6980   115800    68200   184000
      31   314500     9800     6783   118300    64200   182400
      32   304200    10300     6578   120200    60200   180400
      33   293500    10700     6363   121600    56200   177800
      34   282400    11100     6139   122300    52300   174700
      35   270900    11500     5907   122500    48500   171000
      36   259000    11900     5666   122000    44800   166800
      37   246700    12300     5417   120900    41100   162000
      38   234000    12700     5160   119000    37600   156600
      39   221000    13000     4894   116400    34200   150600
      40   207700    13300     4623   113200    30900   144100
      41   194200    13500     4344   109400    27700   137100
      42   180600    13600     4062   104900    24700   129700
      43   166900    13700     3778    99900    21900   121800
      44   153200    13700     3491    94400    19200   113600
      45   139600    13600     3204    88400    16700   105100
        Table II (Continued)


           Age 30                     Reserves Asset    Liability
 Business Insurance         Premium  4 1/2%   NPV      NPV
  Year     Inforce  Claims   Income   30% Load Premiums Benefits

      46   126300    13300     2920    82100    14500    96500
      47   113300    13000     2642    75500    12300    87900
      48   100700    12600     2370    68800    10400    79200
      49    88600    12100     2106    62000     8700    70700
      50    77100    11500     1853    55200     7200    62400
      51    66300    10800     1613    48600     5800    54400
      52    56300    10000     1387    42200     4600    46800
      53    47100     9200     1178    36100     3600    39700
      54    38800     8300      985    30500     2700    33200
      55    31400     7400      812    25300     2000    27300
      56    25000     6400      657    20700     1400    22100
      57    19400     5600      523    16600      900    17500
      58    14700     4700      406    13100      500    13600
      59    10800     3900      307    10100      200    10300
      60        0    10800      226        0        0        0


        Table III
Overview of the business outcomes which result when a life
insurance bank insures a cohort of 40 year olds, each one for the
same sum, in a total of $750,000.  The annual premium is
percent.

                                     Life Reserves on the Basis
  Business Insurance         Premium  3 1/2 % Interest &
  Year     Inforce  Claims   Income   I = 0%   I = 1%   I =1 1/4% -
- --- ---- --- ---- --- ---- --- ---- --- ---- --- ---- --- ---
     0   750000
       1   742200     7800    22062    11500     4204     2378
       2   734300     7900    21833    23100    16008    14230
       3   726300     8000    21601    34800    27894    26165
       4   718100     8200    21365    46500    39812    38133
       5   709700     8400    21124    58200    51704    50076
       6   701000     8700    20877    69800    63522    61944
       7   692000     9000    20621    81300    75194    73667
       8   682600     9400    20356    92600    86679    85204
       9   672900     9700    20080   103600    97949    96526
      10   662800    10100    19794   114400   108957   107586
      11   652200    10600    19497   124900   119644   118326
      12   641200    11000    19186   135000   129979   128714
      13   629700    11500    18862   144700   139897   138685
      14   617700    12000    18524   154000   149349   148190
      15   605200    12500    18171   162700   158298   157191
      16   592100    13100    17803   170900   166658   165605
      17   578400    13700    17418   178400   174378   173378
      18   564100    14300    17015   185200   181421   180474
      19   549200    14900    16594   191300   187727   186833
      20   533700    15500    16156   196600   193243   192401
      21   517500    16200    15700   201000   197859   197068
      22   500600    16900    15223   204500   201517   200777
      23   483000    17600    14726   206900   204156   203466
      24   464700    18300    14208   208300   205726   205085
      25   445700    19000    13670   208500   206172   205579
      26   426100    19600    13111   207700   205493   204947
      27   405800    20300    12534   205600   203605   203105
      28   384900    20900    11937   202300   200517   200061
      29   363500    21400    11322   197900   196260   195846
      30   341700    21800    10693   192400   190877   190503
      31   319500    22200    10052   185700   184356   184021
      32   297100    22400     9399   178000   176810   176513
      33   274600    22500     8740   169400   168309   168046
      34   252100    22500     8078   159900   158933   158702
      35   229800    22300     7416   149600   148829   148629
      36   207800    22000     6760   138800   138099   137927
      37   186400    21400     6113   127600   126989   126842
      38   165600    20800     5483   116000   115546   115422
      39   145700    19900     4871   104400   104035   103932
      40   126700    19000     4286    92900    92524    92440
        Table III (Continued)

                                     Life Reserves on the Basis
  Business Insurance         Premium  3 1/2 % Interest &
  Year     Inforce  Claims   Income   I = 0%   I = 1%   I =1 1/4% -
- --- ---- --- ---- --- ---- --- ---- --- ---- --- ---- --- ---
      41   108900    17800     3727    81600    81300    81231
      42    92400    16500     3203    70700    70513    70459
      43    77300    15100     2718    60500    60318    60276
      44    63700    13600     2274    51000    50863    50832
      45    51600    12100     1874    42300    42223    42200
      46    41000    10600     1518    34500    34462    34446
      47    31900     9100     1206    27700    27644    27634
      48    24200     7700      938    21800    21749    21742
      49    17800     6400      712    16800    16745    16743
      50        0     5200        0     6400        0        0

        Table III (Continued)

         Age 40                     Reserves Asset    Liability
Business Insurance         Premium  4 1/2%   NPV      NPV
Year     Inforce  Claims   Income   30% Load Premiums Benefits
     0   750000
     1   742200     7800    22062   -65700   327800   262400
     2   734300     7900    21833   -53700   319700   266300
     3   726300     8000    21601   -41600   311500   270300
     4   718100     8200    21365   -29200   303200   274300
     5   709700     8400    21124   -16900   294800   278200
     6   701000     8700    20877    -4500   286200   282000
     7   692000     9000    20621     7900   277500   285700
     8   682600     9400    20356    20200   268700   289200
     9   672900     9700    20080    32400   259900   292500
    10   662800    10100    19794    44400   250900   295500
    11   652200    10600    19497    56200   241800   298300
    12   641200    11000    19186    67800   232600   300700
    13   629700    11500    18862    79100   223400   302700
    14   617700    12000    18524    90000   214100   304300
    15   605200    12500    18171   100500   204700   305500
    16   592100    13100    17803   110600   195400   306100
    17   578400    13700    17418   120100   185900   306200
    18   564100    14300    17015   129000   176500   305700
    19   549200    14900    16594   137300   167100   304600
    20   533700    15500    16156   144800   157800   302800
    21   517500    16200    15700   151600   148500   300200
    22   500600    16900    15223   157400   139200   296800
    23   483000    17600    14726   162300   130100   292600
    24   464700    18300    14208   166200   121100   287400
    25   445700    19000    13670   169000   112200   281400
    26   426100    19600    13111   170700   103600   274400
    27   405800    20300    12534   171200    95200   266500
    28   384900    20900    11937   170500    87000   257600
    29   363500    21400    11322   168600    79100   247800
    30   341700    21800    10693   165600    71500   237100
    31   319500    22200    10052   161400    64200   225600
    32   297100    22400     9399   156000    57200   213300
    33   274600    22500     8740   149700    50700   200400
    34   252100    22500     8078   142400    44500   186900
    35   229800    22300     7416   134200    38800   173100
    36   207800    22000     6760   125400    33400   158800
    37   186400    21400     6113   116000    28600   144600
    38   165600    20800     5483   106200    24100   130300
    39   145700    19900     4871    96100    20100   116300
    40   126700    19000     4286    85900    16500   102500
    41   108900    17800     3727    75900    13400    89300
    42    92400    16500     3203    66200    10600    76800
    43    77300    15100     2718    56900     8300    65200
    44    63700    13600     2274    48200     6300    54500
    45    51600    12100     1874    40300     4600    44900

        Table III (Continued)

         Age 40                     Reserves Asset    Liability
Business Insurance         Premium  4 1/2%   NPV      NPV
Year     Inforce  Claims   Income   30% Load Premiums Benefits

    46    41000    10600     1518    33100     3200    36300
    47    31900     9100     1206    26700     2100    28800
    48    24200     7700      938    21200     1200    22400
    49    17800     6400      712    16500      500    17000
    50        0    17800      524        0        0        0


        Table IV
Overview of the business outcomes which result when a life
insurance bank insures a cohort of  50 year olds, each one for the
same sum, in a total of $250,000.  The annual premium is
percent.
                                     Life Reserves on the Basis
 Business Insurance         Premium  3 1/2 % Interest &
 Year     Inforce  Claims   Income   I = 0%   I = 1%   I =1 1/4%
 - --- ---- --- ---- --- ---- --- ---- --- ---- --- ---- --- ---
        0   250000
        1   246000     4000    11271     5600     3150     2605
        2   241800     4200    11090    11100     8728     8206
        3   237500     4300    10901    16400    14159    13658
        4   233000     4500    10707    21600    19417    18937
        5   228300     4700    10504    26500    24484    24027
        6   223400     4900    10293    31300    29335    28900
        7   218200     5200    10072    35800    33932    33519
        8   212800     5400     9837    40000    38271    37879
        9   207200     5600     9594    44000    42326    41956
       10   201300     5900     9341    47600    46052    45704
       11   195200     6100     9075    50900    49436    49109
       12   188800     6400     8800    53800    52430    52124
       13   182200     6600     8512    56300    55031    54746
       14   175300     6900     8214    58400    57191    56926
       15   168100     7200     7903    60000    58881    58636
       16   160700     7400     7579    61100    60113    59888
       17   153000     7700     7245    61800    60835    60628
       18   145100     7900     6898    61900    61065    60877
       19   137000     8100     6542    61600    60796    60625
       20   128800     8200     6176    60800    60068    59914
       21   120400     8400     5807    59500    58830    58692
       22   112000     8400     5428    57700    57177    57054
       23   103500     8500     5049    55600    55066    54957
       24    95000     8500     4666    53000    52554    52459
       25    86600     8400     4283    50100    49709    49627
       26    78300     8300     3904    46900    46548    46477
       27    70200     8100     3530    43400    43149    43089
       28    62400     7800     3165    39800    39597    39546
       29    54900     7500     2813    36100    35920    35877
       30    47800     7100     2475    32400    32212    32177
       31    41100     6700     2155    28600    28507    28478
       32    34900     6200     1853    25000    24906    24883
       33    29200     5700     1573    21500    21444    21427
       34    24100     5100     1316    18300    18229    18215
       35    19500     4600     1087    15300    15216    15206
       36    15500     4000      879    12500    12512    12505
       37    12100     3400      699    10200    10148    10144
       38     9200     2900      546     8100     8073     8071
       39     6800     2400      415     6300     6313     6312
       40        0     6800      307        0        0        0

        Table IV (Continued)

         Age 50                     Reserves Asset    Liability
Business Insurance         Premium  4 1/2%   NPV      NPV
Year     Inforce  Claims   Income   30% Load Premiums Benefits
     0   250000
     1   246000     4000    11271   -27200   139800   112500
     2   241800     4200    11090   -21000   134400   113400
     3   237500     4300    10901   -14900   129100   114200
     4   233000     4500    10707    -8900   123800   114800
     5   228300     4700    10504    -3000   118400   115200
     6   223400     4900    10293     2600   113000   115500
     7   218200     5200    10072     8100   107500   115500
     8   212800     5400     9837    13300   102100   115300
     9   207200     5600     9594    18300    96600   114900
    10   201300     5900     9341    23100    91200   114200
    11   195200     6100     9075    27500    85800   113200
    12   188800     6400     8800    31500    80500   111900
    13   182200     6600     8512    35200    75200   110400
    14   175300     6900     8214    38500    70000   108400
    15   168100     7200     7903    41300    64900   106100
    16   160700     7400     7579    43700    59900   103500
    17   153000     7700     7245    45500    55000   100500
    18   145100     7900     6898    46900    50300    97100
    19   137000     8100     6542    47700    45700    93400
    20   128800     8200     6176    48100    41300    89400
    21   120400     8400     5807    48000    37100    85000
    22   112000     8400     5428    47400    33100    80400
    23   103500     8500     5049    46300    29300    75500
    24    95000     8500     4666    44800    25700    70400
    25    86600     8400     4283    42800    22400    65200
    26    78300     8300     3904    40600    19300    59900
    27    70200     8100     3530    38000    16500    54500
    28    62400     7800     3165    35200    13900    49100
    29    54900     7500     2813    32200    11600    43800
    30    47800     7100     2475    29100     9600    38700
    31    41100     6700     2155    26000     7700    33700
    32    34900     6200     1853    22900     6200    29000
    33    29200     5700     1573    19800     4800    24600
    34    24100     5100     1316    17000     3600    20600
    35    19500     4600     1087    14300     2700    17000
    36    15500     4000      879    11900     1900    13700
    37    12100     3400      699     9700     1200    10900
    38     9200     2900      546     7800      700     8500
    39     6800     2400      415     6200      300     6500
    40        0     6800      307        0        0        0


        Table V
Overview of the business results achieved by a life insurance bank
when it closes annually insurance policies in the amount of $1.5
million [There are annually insured 30 year olds with $500,000, 40
year olds with $750,000, and 50 year olds with $250,000].

Business Insurance         Premium  Life Reserves: Closing Exp
Year     Inforce  Claims   Income   0 %      1 %      1 1/4 %

      1  1484000    16000    43800    22200     7500     3900
      2  2951600    32400    87100    66600    37700    30600
      3  4402600    49000   129900   133200    90500    80000
      4  5836600    66000   172100   222000   165800   151900
      5  7253100    83500   213900   332700   263400   246400
      6  8651600   101500   255000   465200   383300   363100
      7 10031400   120200   295600   619100   524900   501700
      8 11391900   139500   335700   794000   687900   661700
      9 12732500   159400   375100   989400   871800   842900
     10 14052400   180100   413800  1204800  1076200  1044500
     11 15350900   201500   451900  1439600  1300300  1266000
     12 16627200   223700   489400  1693100  1543600  1506800
     13 17880500   246700   526100  1964500  1805200  1765900
     14 19109900   270600   562000  2253000  2084200  2042600
     15 20314500   295400   597200  2557400  2379600  2335800
     16 21493300   321200   631600  2876800  2690300  2644400
     17 22645200   348100   665200  3209900  3015100  2967200
     18 23769200   376000   697900  3555200  3352700  3302800
     19 24864300   404900   729700  3911600  3701700  3650000
     20 25929500   434800   760600  4277500  4060600  4007200
     21 26963700   465800   790600  4651400  4427900  4372800
     22 27965900   497800   819500  5031700  4801900  4745300
     23 28935000   530900   847400  5416700  5181000  5122900
     24 29870000   565000   874300  5804600  5563300  5503900
     25 30770000   600000   900100  6193500  5947200  5886500
     26 31634100   635900   924800  6581900  6330700  6268900
     27 32461500   672600   948400  6967700  6712200  6649200
     28 33251500   710000   970800  7349300  7089700  7025700
     29 34003600   747900   992100  7725000  7461700  7396800
     30 34717400   786200  1012300  8093300  7826400  7760600
     31 35392500   824900  1031300  8452300  8182300  8115700
     32 36028700   863800  1049100  8800600  8527700  8460500
     33 36626000   902700  1065800  9136900  8861400  8793500
     34 37184600   941400  1081300  9459900  9182000  9113500
     35 37704800   979800  1095700  9768400  9488400  9419400
     36 38187100  1017700  1109000 10061500  9779600  9710100
     37 38632300  1054800  1121300 10338500 10054900  9985000
     38 39041100  1091200  1132500 10598600 10313600 10243300
     39 39414600  1126500  1142600 10841400 10555100 10484500
     40 39749000  1165600  1151900 11061900 10774400 10703500

        Table V (Continued)
Business Insurance         Premium  Life Reserves: Closing Exp
Year     Inforce  Claims   Income   0 %      1 %      1 1/4 %

     41 40052100  1196900  1159900 11265900 10977400 10906200
     42 40325100  1227000  1167200 11453300 11164000 11092600
     43 40569300  1255800  1173700 11624200 11334100 11262600
     44 40786200  1283100  1179400 11778900 11488200 11416500
     45 40977400  1308800  1184500 11917800 11626600 11554800
     46 41144700  1332700  1189000 12041600 11750000 11678000
     47 41289900  1354800  1192800 12151000 11859000 11787000
     48 41414800  1375100  1196100 12246900 11954600 11882500
     49 41521200  1393600  1198900 12330200 12037600 11965400
     50 41598300  1410300  1200800 12395500 12096300 12024100
     51 41664600  1421100  1202400 12447100 12147800 12075500
     52 41720900  1431100  1203800 12491700 12192300 12120000
     53 41768000  1440300  1205000 12529800 12230300 12158000
     54 41806800  1448600  1206000 12561800 12262200 12189900
     55 41838200  1456000  1206800 12588200 12288500 12216200
     56 41863200  1462400  1207400 12609700 12310000 12237700
     57 41882600  1468000  1207900 12626800 12327100 12254800
     58 41897300  1472700  1208400 12640200 12340500 12268200
     59 41908100  1476600  1208700 12650400 12350800 12278400
     60 41908100  1487400  1208900 12650400 12350800 12278400


        Table V (Continued)

                      Reserves
    Business Insurance@4 1/2%
    Year     Inforce  Load 30% NSP

           1  1484000  -128800
           2  2951600  -234500
           3  4402600  -316900
           4  5836600  -375800
           5  7253100  -411200
           6  8651600  -423300
           7 10031400  -412100
           8 11391900  -377900
           9 12732500  -320900
          10 14052400  -241400
          11 15350900  -139900
          12 16627200   -17000
          13 17880500   126900
          14 19109900   291000
          15 20314500   474400
          16 21493300   676400
          17 22645200   895700
          18 23769200  1131200
          19 24864300  1381600
          20 25929500  1645700
          21 26963700  1922000
          22 27965900  2209000
          23 28935000  2505000
          24 29870000  2808400
          25 30770000  3117300
          26 31634100  3430200
          27 32461500  3745100
          28 33251500  4060200
          29 34003600  4373800
          30 34717400  4684300
          31 35392500  4990000
          32 36028700  5289100
          33 36626000  5580200
          34 37184600  5861900
          35 37704800  6132900
          36 38187100  6392200
          37 38632300  6638800
          38 39041100  6871800
          39 39414600  7090500
          40 39749000  7289600



                                Table V (Continued)

                      Reserves
    Business Insurance@4 1/2%
    Year     Inforce  Load 30% NSP

          41 40052100  7474900
          42 40325100  7646000
          43 40569300  7802800
          44 40786200  7945400
          45 40977400  8074100
          46 41144700  8189300
          47 41289900  8291500
          48 41414800  8381500
          49 41521200  8460000
          50 41598300  8515200
          51 41664600  8563800
          52 41720900  8606000
          53 41768000  8642100
          54 41806800  8672600
          55 41838200  8697900
          56 41863200  8718600
          57 41882600  8735200
          58 41897300  8748300
          59 41908100  8758400
          60 41908100  8758400


        Table VI
Overview of the business results when a life insurance bank closes
insurance policies in the amount of $1.5 million in the first year
and in each of the following years $150,000 more than in the
preceding year [The new insurance polices are distributed over the
ages 30, 40, and 50 in the same relationship as       Table V].


    Business InsurancePremium  Life Reserves:  Closing Cost
    Year     Inforce  Income   0 %      1 %      1 1/4 %
           0  1500000
           1  1484000    43800    22200     7500     3900
           2  3100000    91500    68800    38500    31000
           3  4846200   143000   142100    95000    83500
           4  6720400   198200   244200   179400   163300
           5  8720600   257200   377100   293500   273000
           6 10844400   319700   542900   439800   414400
           7 13089400   385700   743300   619700   589300
           8 15453000   455500   980100   835200   799400
           9 17932800   528400  1254900  1087900  1046900
          10 20525900   604600  1569200  1379500  1332700
          11 23229700   684100  1924500  1711200  1658600
          12 26041100   766800  2322000  2084500  2026100
          13 28957100   852500  2762700  2500500  2435800
          14 31974500   940900  3247600  2960000  2889100
          15 35090100  1032300  3777300  3463800  3386600
          16 38300400  1126400  4352500  4012400  3928700
          17 41601600  1223200  4973300  4606200  4516000
          18 44990100  1322400  5639600  5245400  5148300
          19 48462100  1424000  6351500  5929800  5825900
          20 52013800  1527900  7108500  6658800  6548100
          21 55640900  1634100  7910200  7432200  7314400
          22 59339500  1741900  8755600  8248900  8124100
          23 63105200  1851700  9643800  9108300  8976200
          24 66933700  1963400 10573400 10008600  9869500
          25 70820700  2076700 11542700 10948900 10802600
          26 74761800  2191400 12550500 11927000 11773700
          27 78752600  2307500 13594500 12941700 12780800
          28 82788700  2424600 14672800 13990300 13822100
          29 86866000  2542900 15783500 15071400 14896000
          30 90980200  2662500 16924300 16182100 15999200


        Table VI (Continued)

                       Reserve
    Business Insurance 4.5%
    Year     Inforce   Load 30% NSP

           1  1484000  -128800
           2  3100000  -247400
           3  4846200  -353200
           4  6720400  -443800
           5  8720600  -516800
           6 10844400  -570000
           7 13089400  -601200
           8 15453000  -608200
           9 17932800  -589000
          10 20525900  -541500
          11 23229700  -464200
          12 26041100  -355300
          13 28957100  -213100
          14 31974500   -36300
          15 35090100   176200
          16 38300400   425700
          17 41601600   712600
          18 44990100  1037700
          19 48462100  1401200
          20 52013800  1803500
          21 55640900  2244300
          22 59339500  2723500
          23 63105200  3240400
          24 66933700  3794300
          25 70820700  4384100
          26 74761800  5008700
          27 78752600  5666600
          28 82788700  6356200
          29 86866000  7075800
          30 90980200  7823700


The introduction of a closing commission recommends itself on other grounds also. For example, how is an agent to be repaid who in a short time has made significant sales when he is forced (by death or other conditions) to give up the business? If we seek to rationally establish the amortization of the account of the prepaid commission, especially in a way that is free from the disadvantages of the second and third objection then automatically one arrives at the procedure proposed by us.

We intend here for a partial premium which keeps the same value for each issue age, which is a different fraction of the premium for each issue age, and which is a constant percent of the insured sum. If the amount by which the first annual premium shall be smaller than the following annual premium is a percentage of the premium, the formula for the calculation changes somewhat. In such a case, the following considerations do not fit exactly. Since these maximal rates grow with the issue age in a similar relationship as the premium, it seems preferable to measure the closing costs in percent of the first annual premium. The results of that calculation are shown in the table above. It is evident that here the maximal percentage selected is not exposed to as great a fluctuation.

If the closing costs are calculated in percent of the first annual premium, the agent will also receive a closing commission as a percent of the first annual premium. He would naturally prefer to direct his efforts to the attraction of older persons to insurance. If at the end of the business year an insurance policy has not been in existence for a full insurance year, the reserve for the end of the business year naturally has to be calculated according to the calculations of fractional duration reserve. If, for example, we calculate the reserves as if the full annual premiums had been paid at the beginning of each insurance year for such an insurance policy for which the reserve would be negative or rather zero at the end of the insurance year we must reserve enough so that the calculable mortality for the not yet elapsed fraction of the first insurance year is covered.

Aside from the other consequences that such figures must have on the financial development, this procedure seems to be unjustified, not to use a stronger expression, because this profit can only be realized when the insureds maintain their insurance inforce until the regular expiration or even could be forced to maintain them inforce. But the bank knows that a number of the insurance policies will lapse and in spite of this ascribes the profit of these to itself. It is really very mildly judged when one asks as already Mr. Lazarus did in his essay "What would one say when a merchant puts as an increase of his assets profit which is hoped for out of an initiated, but unconsummated deal?" However the merchant could achieve this goal while the insurance bank knows in advance that it will not achieve its goal in so and so many cases.

Here the profit is greater than the premium income. It generally happens often with the companies that calculate in this way that they have more profit than they own in cash or money value. Therefore one frequently finds that colossal profit postings have been held back for later distribution. This gain is by as much higher the higher the loading -- Therefore one finds in England almost only mutual or mixed companies. Pure stock companies would have o add the possibly smallest loading to their net premiums.

                                                            School of Business Home Page  |  Washburn Home Page